Optical solitons in a fractional cubic–quintic Schrödinger equation: traveling wave and modulation instability

Ziyad A. Alhussain; Ziyad A. Alhussain

doi:10.3934/math.20261229

AIMS Mathematics

2026, Volume 11, Issue 9: 31062-31112. doi: 10.3934/math.20261229

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Research article

Optical solitons in a fractional cubic–quintic Schrödinger equation: traveling wave and modulation instability

Ziyad A. Alhussain ^,

Department of Statistics and Operations Research, College of Science, Qassim University, Saudi Arabia

Received: 18 May 2026 Revised: 06 September 2026 Accepted: 14 September 2026 Published: 22 September 2026
MSC : 34A08, 35C08, 35Q55, 35R11, 37K40

In the present work, the conformable fractional cubic–quintic nonlinear Schrödinger equation is investigated through traveling-wave analysis, nonlinear dynamics, and stability characterization of optical wave solutions. By applying a suitable traveling-wave transformation, the governing conformable fractional model is reduced to a nonlinear ordinary differential equation. Several analytical approaches, including the variational method, the projective Riccati equation method, and the $ \left(\mathscr{G}'/\mathscr{G}^{{2}}\right) $-expansion method, are employed to construct different classes of optical wave profiles, distinguished throughout as exact, piecewise classical, or variational. The resulting profiles describe various nonlinear wave structures, including bright, dark, periodic, singular, and localized optical waves. Furthermore, the associated reduced dynamics are analyzed through phase portraits and bifurcation diagrams, while the wave profiles illustrate the influence of the governing parameters on nonlinear propagation. Modulational instability of the continuous-wave background and spectral stability of the traveling-wave families for which a spectral problem on the line is defined are analyzed to identify parameter regimes associated with perturbation growth. Two independent discretization schemes agree to their truncation error, and the numerical benchmark reproduces the predicted growth rate to within about $ 2\% $. Grid refinement reveals numerically neutral modes that can appear weakly unstable on a fixed grid, and the apparent ordering of perturbation amplification with the conformable order depends on the observation window rather than on the intrinsic growth rate. The cubic–quintic nonlinearities thus govern the intrinsic instability, while the conformable order affects the temporal representation and finite window observation of the waves.
- fractional derivatives,
- Schrödinger equation,
- cubic–quintic nonlinearity,
- conformable derivative,
- optical solitons,
- stability analysis
Citation: Ziyad A. Alhussain. Optical solitons in a fractional cubic–quintic Schrödinger equation: traveling wave and modulation instability[J]. AIMS Mathematics, 2026, 11(9): 31062-31112. doi: 10.3934/math.20261229

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Abstract

In the present work, the conformable fractional cubic–quintic nonlinear Schrödinger equation is investigated through traveling-wave analysis, nonlinear dynamics, and stability characterization of optical wave solutions. By applying a suitable traveling-wave transformation, the governing conformable fractional model is reduced to a nonlinear ordinary differential equation. Several analytical approaches, including the variational method, the projective Riccati equation method, and the $ \left(\mathscr{G}'/\mathscr{G}^{{2}}\right) $-expansion method, are employed to construct different classes of optical wave profiles, distinguished throughout as exact, piecewise classical, or variational. The resulting profiles describe various nonlinear wave structures, including bright, dark, periodic, singular, and localized optical waves. Furthermore, the associated reduced dynamics are analyzed through phase portraits and bifurcation diagrams, while the wave profiles illustrate the influence of the governing parameters on nonlinear propagation. Modulational instability of the continuous-wave background and spectral stability of the traveling-wave families for which a spectral problem on the line is defined are analyzed to identify parameter regimes associated with perturbation growth. Two independent discretization schemes agree to their truncation error, and the numerical benchmark reproduces the predicted growth rate to within about $ 2\% $. Grid refinement reveals numerically neutral modes that can appear weakly unstable on a fixed grid, and the apparent ordering of perturbation amplification with the conformable order depends on the observation window rather than on the intrinsic growth rate. The cubic–quintic nonlinearities thus govern the intrinsic instability, while the conformable order affects the temporal representation and finite window observation of the waves.

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